All Targets Near and Far POGIL ACTIVITY.3
Name ______
POGIL ACTIVITY 3 All Targets Near and Far: Accuracy, Precision and Significant Figures
Accuracy and Precision
ll measurements are estimates of some true value. The validity of any A measurement is primarily due to the nature of the device used to make the measurement and partly due to the manner in which data is collected.
Model 1 Accuracy and Precision low accuracy low accuracy high accuracy high accuracy low precision high precision low precision high precision
A measured value that matches the true value is a "Bull's-eye".
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Critical Thinking Questions CTQ 1 Using Model 1, define accuracy using a grammatically correct sentence.
CTQ 2 Define precision using a grammatically correct sentence.
CTQ 3 High precision implies high accuracy, true or false?
CTQ 4 High accuracy implies high precision, true or false?
ccording to the U.S. Mint, the mass of a U.S. quarter ($0.25) is 5.670 g. Four A students measured the same coin (Table 1).
Table 1. Mass of U.S. Quarter I II III IV 5.010 g 5.670 g 5.673 g 5.669 g 5.010 g 5.589 g 5.674 g 5.672 g 5.011 g 5.801 g 5.673 g 5.672 g 5.011 g 5.800 g 5.669 g 5.671 g 5.010 g 5.712 g 5.669 g 5.669 g
CTQ 5 Which data set is the most precise? I II III IV
CTQ 6 What information is required to determine the accuracy of any data set?
CTQ 7 Explain how you could determine (mathematically) which set is the most accurate?
CTQ 8 Which data set is the most accurate? I II III IV
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Name ______B. Uncertainty (or Error) very measurement is an estimate of the actual value because every measurement E contains a degree of uncertainty or error. In this context, error does not mean “mistake”. Error (uncertainty) is a mathematically determined variance between individual measurements that arises when repeated measurements are made on the same sample or object. For example, four students weighed a U.S. dime ($0.10) multiple times using different types of balances (Table 2).
Table 2. Mass of U.S. Dime I II III IV 2.2 g 2.29 g 2.268 g 2.2681 g 2.1 g 2.26 g 2.269 g 2.2689 g 2.2 g 2.21 g 2.260 g 2.2683 g 2.3 g 2.23 g 2.267 g 2.2682 g 2.3 g 2.27 g 2.268 g 2.2680 g
The measurements in Set I contain two digits. The first digit (the one’s place) is called the reproducible digit and the second digit in the tenth’s place (0.1) is called the doubtful digit.
CTQ 9 Formulate a definition for the term reproducible digit.
CTQ 10 Formulate a definition for the term doubtful digit.
CTQ 11 Identify the doubtful digit in each data set in Table 2.
The doubtful digit is: data the one’s one-tenth’s hundredth’s thousandth’s ten-thousandth’s set place place place place place I II III IV
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CTQ 12 What is the uncertainty in each data set from Table 2?
Data Set +/- 0.1 g +/- 0.01 g +/- 0.001 g +/- 0.0001 g I II III IV
CTQ 13 An increase in precision is: a. an increase in uncertainty b. a decrease in uncertainty
C. Significant Figures (significant digits, sig figs), Exact Numbers and Those Pesky Zeros
ignificant figures are those digits in any number that designate the level of Sprecision. Sig figs include all reproducible digits plus the first doubtful digit.
Table 3. Mass of U.S. Nickel I II III IV 5.0 g 5.01 g 5.001 g 5.0001 g 5.0 g 5.00 g 5.002 g 5.0009g 5.0 g 5.01 g 5.002 g 5.0002 g 5.1 g 5.01 g 5.000 g 5.0006 g 5.0 g 5.02 g 5.002 g 5.0003 g
CTQ 14 According to the definition above, indicate how many sig figs are in each measurement: a. Set I _____ b. Set II _____ c. Set III _____ d. Set IV _____
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xact numbers contain no error (uncertainty) since these numbers are not E measurements; in other words, an exact count is not an estimate. For example, the number of people in a family of four is an exact number (Table 4).
Table 4. Measurements versus Exact Numbers Measurements (estimates) Exact Numbers 24 students in 5 fingers 0.00024 kg U.S. population classroom each hand volume of a 1-L 17 one-liter 2 cups of milk 1 dozen = 12 water bottle water bottles 18 mph 9.3 x 106 miles 2.54 cm = 1 inch 100 cm = 1 meter
on-zero digits are always significant but zeros may or may not be significant. In N fact, zeros cause the most confusion when dealing with significant digits. Examine Table 5.
Table 5. Zeros are not always significant 1 sig fig 2 sig fig 3 sig fig 4 sig fig 5 sig fig 3 kg 2.9 kg 2.98 kg 2.982 kg 2.9824 kg 80 km 81 km 80.7 km 80.76 km 80.761 km 0.06 g 0.055 g 0.0550 g 0.05503 g 0.055039 g 200 L 2.0 20.0 200.0 L 200.00 L 3 x 108 m 3.0 x 108 m 3.00 x 108 m 3.008 x 108 m 3.0081 x 108 m
CTQ 15 Using Table 5 and the examples shown here, answer true or false: examples T F Zeros between non-zero digits 80.7 are significant 3.008 x 108 Zeros at the beginning of a number 0.06 are significant: 0.055 Zeros in the coefficient for scientific notation 3.0 x 108 are significant 1.00 x 10-3 Trailing zeros in numbers written with a decimal point 200.0 are significant 8.750 Trailing zeros in numbers written without a decimal point 80 are significant 200
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D. Calculations Involving Sig Figs and Rounding Numbers
calculation cannot change the level of uncertainty. There are two rules to observe, A one for multiplication and one for addition:
Rule 1 When multiplying (or dividing) measurements, the answer must contain the same number of sig figs as the starting quantity with the fewest sig figs.
For example, calculate the mileage for an automobile based on the following data, 108.4 miles driven using 3.5 gallons of fuel:
108.4 mi = 30.97142857 mpg = 31 mpg 3.5 gal Since the denominator has fewer sig figs than the numerator, the answer must be rounded to two sig figs.
Rule 2 When adding (or subtracting) measurements, the answer must contain the same number of decimal places as the starting quantity with the fewest decimal places.
Consider the addition of volume in this example: 7.6 mL + 125 mL = 132.6 mL = 133 mL count decimal places, not sig figs In this case, one measurement (125 mL) does not have any digits past the one’s place and the answer must be rounded to the nearest one milliliter.
hen calculations generate trailing zeros, conversion to scientific notation clarifies the Wnumber of sig figs allowed in the answer:
730 cm + 880 cm = 1610 cm = 1.610 x 103 cm Trailing zero is allowed since precision is to the nearest one’s place.
525 g x 4 = 2100 g = 2 x 103 g Trailing zeros are not allowed because the precision is limited to one sig fig.
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ounding Numbers: Calculations often generate non-significant digits, especially Rcalculations done with an electronic calculator; these values must be rounded to the correct number of sig figs. Once it is determined how many sig figs are allowed, all remaining digits to the right must be dropped. However, if the first digit to be dropped is 5 or greater, then the preceding digit is rounded up by one, then the extra digits are dropped.
In these examples, two sig figs are allowed in the answer: 2.1 x 3.5 = 7.35 = 7.4 round up the doubtful digit by one then truncate extra digit 2.1 x 3.4 = 7.14 = 7.1 do not round up, truncate extra digit
In this example, three sig figs are allowed in the answer: 252 / 1.401 = 179.8715203 = 180 = 1.80 x 102 round up and convert to sci. notation Notice the zero generated after rounding- this is the most ambiguous type of zero, a trailing zero with no decimal point. But inspection of the calculation reveals this to be a sig fig and should be included in the final answer. Conversion to scientific notation removes all doubt that this zero is significant and should be included in the final answer.
ore about zeros. Frequently, rounding off numbers produces zeros which may Mor may not be significant. Scientific notation is the best method to remove all ambiguity and to eliminate placeholder zeros. Below are four more examples of how to deal with zeros.
1. Zeros generated to the left of the decimal point after rounding off become placeholder zeros (to hold the decimal point) but are not significant digits.
rounded to 3 digits 3 sig figs retained 8 0 0 1 9 8 0 0 0 0 8 . 0 0 x 104 replace last unambigous do not two digits with notation round up placeholder zeros to hold this zero zeros decimal point
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2. Zeros generated to the right of the decimal point after rounding off may be significant depending on how many digits are allowed in the final answer.
rounded to 3 sig figs 0 . 0 0 9 5 9 7 0 . 0 0 9 6 0 9 . 6 0 x 10-3 round up sometimes rouding up affects two or the 9 truncate more digits! the 7
3. Beware of Zeros that “disappear” from electronic calculators. Compare the addition problem below done by hand versus using a calculator.
Addition by pencil: Addition by calculator: 87.72 + 2.28 = 90.00 (keep 2 decimal places) 87.72 + 2.28 = 90
4. Always include a “courtesy zero” for numbers that begin with a decimal point. Your instructor will love you for this!
Courtesy zeros are not significant digits.
Examples: (0.01 not .01) (0.005 not .005) (0.100 not .100 and not .1)
Exercises With Sig Figs and Rounding 1. Assume each number is a measured value. How many sig figs are in each number? ___ a. 93 million ___ g. 300 ___ m. 1.81 x 10-5 ___ b. 93,000,000 ___ h. 300.0 ___ n. 6.0 x 100 ___ c. 9.30 x 107 ___ i. 0.607 ___ o. 0.00023 ___ d. 93,125,479 ___ j. 1.277 ___ p. 23,004 ___ e. 0.677 ___ k. 6089 ___ q. 873.20 ___ f. 0.0677 ___ l. 6.700 x 10-2 ___ r. 0.00500
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2. Round each number to 3 sig figs; write answers as decimal numbers and in scientific notation.
decimal value scientific notation a. 30,010 ______b. 30.54 ______c. 609.7 ______d. 5.2377 ______e. 0.1278 ______
3. Solve each calculation and round the answer to the correct number of sig figs. Use good judgment and convert to scientific notation where appropriate- to preserve zeros that are significant or to eliminate placeholder zeros. Include units where indicated. a. 0.433 x 7.6889 b. 7.22 x 19.4
c. 989.17 / 23.1
d. 9.94 + 0.05
e. 6.94 - 0.072
f. 134.9 - 0.05
g. (2.21 + 0.09) x 12.75
h. (2.43 + 0.07) x 34.5
i. (2.5 cm x 0.65 cm)
j. Convert 65.4 g into kg
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NOTES
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